Some faces left behind in pose mode
First of all put the armature modifier before the subsurf and check the two buttons of armature modifier so it acts in edit mode also.

As you can read in the N properties panel, the not moving vertices heve been weighted to the wrong bones (shoulder and upper arm), instead of "spine.006".



Then go to the vertex groups tab, select all wrong moving vertices, remove them from the wrong vertex groups, and assign them to the "spine.006" one.



In general, when an object has to move exactly with one bone, without any deformation, you should avoid automatic weighting, and parent the object directly to the bone (bone option, choose the bone in pose mode), or choose "with empty group" option and manually assign all vertices to the corresponding vertex group.